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The distribution of a random variable whose independent copies span \(\ell _M\) is unique

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Abstract

It is well known that the span in \(L_p\) of a sequence of independent copies of a mean zero random variable \(f\in L_p\) is a subspace isomorphic to some Orlicz sequence space \(\ell _M\). It is also known (Astashkin et al. in Stud Math 230(1):41–57, 2015) that the distribution of such a random variable \(f\in L_p\) is essentially unique. We show that this result continues to hold when \(L_p\) is replaced with an arbitrary symmetric function space and when the Orlicz function M satisfies a natural “submultiplicativity” condition. Thereby we extend the classical results of Kadec and Braverman proved in the case of power functions.

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Authors and Affiliations

  1. Department of Mathematics, Samara National Research University, Moskovskoye Shosse 34, Samara, Russia, 443086

    S. Astashkin

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, 2052, Australia

    F. Sukochev & D. Zanin

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  1. S. Astashkin
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  2. F. Sukochev
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  3. D. Zanin
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Correspondence to S. Astashkin.

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The work of the first named author was completed as a part of the implementation of the development program of the Volga Region Scientific and Educational Mathematical Center (Agreement No. 075-02-2021-1393).

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Astashkin, S., Sukochev, F. & Zanin, D. The distribution of a random variable whose independent copies span \(\ell _M\) is unique. Rev Mat Complut 35, 815–834 (2022). https://doi.org/10.1007/s13163-021-00406-x

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